Donald Trump on Friday sought to distance himself from comments made by Rudy Giuliani about the Stormy Daniels scandal.

The president also reiterated his willingness to speak to special counsel Robert Mueller, who is investigating Russian election interference and links between the Trump campaign and Moscow. Mueller has indicted four former Trump aides.

Giuliani, a former mayor of New York City, joined the president’s legal team last month. In interviews this week, he raised questions about Trump’s own comments about the $130,000 payment that Michael Cohen, Trump’s personal lawyer, paid to Daniels, an actor in adult films.

Speaking to reporters on the south lawn of the White House on Friday morning, Trump said Giuliani “started yesterday … he’ll get his facts straight”.

Cohen has insisted that he paid Daniels, who claims to have had a sexual encounter with Trump, solely from his personal funds and in order to silence her and protect Trump’s reputation on the eve of the 2016 election. Questioned by reporters on Air Force One in April, Trump denied any knowledge of the payments.

In an interview with Fox News host Sean Hannity on Wednesday night, Giuliani contradicted Trump and said the president reimbursed Cohen. Giuliani said the money was “funneled through the law firm and the president repaid it”.

Trump did not contradict Giuliani in a series of tweets on Thursday. In stilted and legalistic prose, he wrote: “Mr Cohen, an attorney, received a monthly retainer, not from the campaign and having nothing to do with the campaign, from which he entered into, through reimbursement, a private contract between two parties, known as a non-disclosure agreement.”

Trump added that “these agreements are … very common among celebrities and people of wealth”.

On Friday, Trump said: “Rudy is a great guy … he’s working hard, learning the subject matter. He is a great guy.”

He added: “He started yesterday. He’ll get his facts straight. It’s actually very simple, there has been a lot of misinformation really.”

Speaking to reporters again by Air Force One at Andrews air force base, Trump said: “When Rudy made the statements … Rudy’s great and Rudy had just started and he wasn’t totally familiar with everything.”

He added: “We love Rudy. He’s a special guy.”

He also insisted he had not changed his story about the payments to Daniels. “Excuse me, excuse me,” Trump said. “You take a look at what I said. You go back and take a look. You’ll see what I said.”

About speaking to Mueller, Trump said: “I would like to speak because we’ve done nothing wrong.” But Trump also noted that his lawyers did not want him to do so, and outlined his oft-repeated belief that the investigation was “a pure witch-hunt” carried out by “all Democrats”.

Giuliani’s comment to Hannity that Trump fired FBI director James Comey because Comey would not confirm that Trump was not the subject of investigation over Russia also caused widespread comment on Wednesday.

Giuliani, Trump said on Friday, “knows it’s a witch-hunt, that’s what he knows. Rudy knows it’s a witch-hunt”.

At the White House, Trump was also asked about progress on his planned meeting with Kim Jong-un, North Korea’s leader.

“We’re having very substantive talks with North Korea and a lot of things have already happened with regard to the hostages,” he said, referring to three Americans held in the country. “I think you’re going to be seeing very good things.”

Trump also appeared to suggest a date and location for his meeting with Kim had been set, and said they would be announced shortly.

The United Nations, meanwhile, said on Friday UN aviation agency directors will visit North Korea next week to discuss a request by the country to open new air routes to South Korea.